Theoretical prediction on the mechanical properties of 3D braided composites using a helix geometry model

Composite Structures 100 (2013) 511–516

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Theoretical prediction on the mechanical properties of 3D braided composites using a helix geometry model Lili Jiang a, Tao Zeng a,⇑, Shi Yan a, Daining Fang b a b

Department of Engineering Mechanics, Harbin University of Science and Technology, Harbin 150080, PR China Department of Mechanics and Aerospace Technology, Peking University, 100871 Beijing, PR China

a r t i c l e

i n f o

Article history: Available online 29 January 2013 Keywords: Helix geometry model 3D braided composites Effective elastic properties Strength

a b s t r a c t In our previous work, we have established a three-dimensional (3D) finite element model (FEM) which precisely simulated the spatial configuration of the braiding yarns. This paper presents a theoretical model based upon the helix geometry unit cell for prediction of the effective elastic constants and the failure strength of 3D braided composites under uniaxial load through the stiffness volume average method and Tsai-Wu polynomial failure criterion. Comparisons between the theoretical and experimental results are conducted. The theoretical results show that the braid angle has significant influences on the mechanical properties of 3D braided composites. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 3D braided composites have been rapidly developed in the past years due to their excellent mechanical properties, such as high specific strength/stiffness, high through-thickness strength and impact resistance. 3D braided composites have been widely used in aerospace, automobile, marine and biomedical, etc. 3D braided composites can be regarded as an assemblage of representative volume element [1,2] that captures the major features of the underlying microstructure and composition in the material. In the recent years, many researchers [3–22] have been devoted to the micro-structures and elastic properties for 3D braided composites. Kregers and Melbardis [3] presented the stiffness volume average method to predict the macroscopic properties of 3D braided composites. Ma et al. [4], Yang et al. [5] and Byun et al. [6] studied the effective elastic properties of 3D braided composites by using ‘Fiber interlock model’, ‘Fiber inclination model’ and ‘fabric geometric model’, respectively. Whitcomb and Woo [7] gave the stress distribution of woven composites using the local finite element method. Wang and Wang [8] reported a mixed volume averaging technique to predict the mechanical behavior of three dimensional braided composites. Wu [9] developed a three-cell model to predict the mechanical properties of 3D braided composites, which can be used to accurately describe the micro-structure. Chen et al. [10] presented a finite multi-phase element model to predict the effective properties of 3D braided composites. Sun and Qiao [11] predicted the strength of 3D braided composites based upon the transverse isotropy of unidirectional laminas. Fang ⇑ Corresponding author. Tel.: +86 451 86390832; fax: +86 451 86390830. E-mail addresses: [email protected], [email protected] (T. Zeng). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.01.016

et al. [12] developed a mesoscopic damage model to study the failure locus of 3D braided four-directional composites under complex loadings. Zeng et al. [13,14] investigated the effective modulus of 3D braided composites with edge and internal cracks. Gu [15] investigated the uniaxial tensile strength of 4-step 3-dimensional braided composites based on the energy conservation, and showed the tensile curve within the whole strain range. Yu and Cui [16] studied the influence of the braiding angle and the fiber volume fraction on the strength for 4-step braided composites. Li and Shen [17] presented thermal postbuckling analysis modeling for 3D braided composite cylindrical shell subjected to a uniform temperature rise. Recently, the finite element methods [18–23] were extensively applied to numerically predict the average stiffness and strength properties of 3D braided composites for their accurate prediction. In our previous work [24], a multiphase finite element method based on the helix geometry model has been presented to predict the effective elastic constants and strength of 3D braided composites under tension loading. The present paper is concerned with the theoretical prediction on the elastic properties and failure strength of 3D braided composites using a helix geometry model. The stiffness property is first compared with test data and the results of the previous micromechanical models. This study is followed by predicting the failure strength of 3D braided composites under axial load. 2. Helix geometry model In our previous work [24], a helix geometry model of 3D braided composites has been presented. A unit cell for the helix geometry model of 3D braided composites is shown in Fig. 1. Four yarns in the helix geometry model are curved to avoid the collision at the

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L. Jiang et al. / Composite Structures 100 (2013) 511–516

Fig. 1. Helix geometry model.

center of the unit-cell, which truly reflects the braided manner and coincides with the actual configuration of the braided composites. Three coordinate systems have been employed in this study, which are: (1) a global coordinate system (x, y, z) with axes aligned parallel to the unit cell axes, (2) a local coordinate system (x0 , y0 , z0 ) with its origin C at the midpoint of the diagonal of the unit cell (x0 -axis: the diagonal direction of unit cell); (3) a local coordinate system (1, 2, 3) with its primary axis parallels to the central axis of the yarn. The local system (1, 2, 3) changes from point to point on a yarn as well as from yarn to yarn. U, V and L refer to the dimensions of the unit cell in x, y and z directions. In order to describe the spatial location of the yarns in the unit cell, the curvature of each yarn should be determined. The center line of the braiding yarns is a parabola defined by the two yarn end points (located on the respective top and bottom surfaces of the unit cell) and the midpoint of the yarn (located on the mid-plane between the top and bottom surfaces), as shown in Fig. 1b. The center line of the braiding yarns in the local coordinate systems (x0 , y0 , z0 ) can be formulated as

(

y0 ¼ c1 þ c2 x0 þ c3 x02 z0 ¼ 0



x ¼ f1 ðzÞ y ¼ f2 ðzÞ

06z6L

ð3Þ

The expressions of f1(z) and h(z) are listed in Appendix A. In Fig. 1c, h(z) is the angle between the tangent of the yarn axis and z-axis and b(z) is the angle between the projection of yarn axis on the xoy plane and x-axis, given by:

1 ffi hðzÞ ¼ arccos rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 dy2 þ þ 1 dz dz bðzÞ ¼ arctan

 dy dx

ð4aÞ

ð4bÞ

ð1Þ 3. Prediction of effective elastic constants

where c1, c2, c3 can be determined by the two end points and the midpoint of the yarn. The center line of the braiding yarns in the global coordinate systems (x, y, z) can be obtained by coordinate transformation

8 09 2 l1 > = 6 y0 ¼ 4 m1 > : 0> ; z n1

where (li, mi, ni) (i = 1, 2, 3) are the direction cosines between the local coordinate system (x0 , y0 , z0 ) and the global coordinate system (x, y, z). Substituting Eq. (2) into Eq. (1), the equation of the center line of the braiding yarns in the global coordinate system can be obtained as

l2 m2 n2

9 38 U > = 7 m3 5 y  V2 > > : ; z  2L n3 l3

ð2Þ

The basic assumption in the present analysis is that the yarns (Fig. 1d) are considered unidirectional composite rods after resin impregnation. A yarn was cut into a few small pieces along the braiding axis z so that the yarns segment in each small piece was assumed to be straight. Each small piece was treated as a transversely isotropic composite with local coordinate system along its yarn segment. The stiffness matrix of each yarn segment in

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L. Jiang et al. / Composite Structures 100 (2013) 511–516

the 1–2–3 coordinate system is expressed as follows as a result of the transverse isotropy:

2

1 E

6 11c 6  12 6 E11 6 c31 6 6 E ½Cs ¼¼ 6 33 6 0 6 6 6 0 4 0

 Ec12 11

 Ec31 33

0

0

1 E11

 Ec31 33

0

0

 Ec31 33

1 E33

0

0

0

0

1 G31

0

0

0

0

1 G31

0

0

0

0

0

3

7 0 7 7 7 0 7 7 7 0 7 7 7 0 7 5

ð5Þ

1 G12

where the Young’s and shear moduli of each yarn segment are obtained from the fiber and matrix properties using micro-mechanics analysis [25]. From the direction cosines between the x–y–z coordinate system and the 1–2–3 coordinate system, the following transformation matrix can be established:

2

2

l1

m21

n21

2m1 n1

2n1 l1

2l1 m1

3

7 6 2 6 l2 m22 n22 2m2 n2 2n2 l2 2l2 m2 7 7 6 7 6 2 2 2 7 6 l3 m n 2m n 2n l 2l m 3 3 3 3 3 3 3 3 ½T ¼ 6 7 6l l m m n n m m þ m n n l þ n l l m þ l m 7 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 27 623 7 6 4 l3 l1 m3 m1 n3 n1 m3 n1 þ m1 n3 n3 l1 þ n1 l3 l3 m1 þ l1 m3 5

Fig. 2. Schematic of a single yarn under tension load.

4. Prediction of strength

l1 l2 m1 m2 n1 n2 m1 n2 þ m2 n1 n1 l2 þ n2 l1 l1 m2 þ l2 m1 ð6Þ where

F p rp þ F pq rp rq ¼ 1 p; q ¼ 1; 2; . . . 6

l1 ¼ cos hðzÞ cos bðzÞ; l2 ¼  sin bðzÞ; l3 ¼ sin hðzÞ cos bðzÞ m1 ¼ cos hðzÞ sin bðzÞ; m2 ¼ cos bðzÞ; m3 ¼ sin hðzÞ sin bðzÞ ð7Þ The stiffness matrix of the unidirectional composite rod is transformed to ½Cs from the local coordinate system (1–2–3) to the global coordinate system (x–y–z):

½Cs ¼ ½T½Cs ½TT

ð8Þ

The effective stiffness matrix of a helix yarn can be obtained by averaging the transformed stiffness matrix of the infinitesimal yarn segment through the length of the yarns.

½Cj ¼

1 l

Z

½T½Cs ½TT ds ðj ¼ 1; 2; 3; 4Þ

ð9Þ

where j denotes the four yarns of a representative unit cell of 3D braided composite. Thus, the stiffness matrix of a unit cell in the global coordinate system can be written as a summation of the four yarns’ stiffness [3]:

½C ¼

4 1X V m ½Cm V m¼1

The longitudinal failure strength of 3D braided composites under uniaxial load is carried out by Tsai-Wu polynomial failure criterion in this analysis. Tsai-Wu second-order tensor polynomial can be written as:

where Fp and Fpq are the second and fourth rank strength tensors. Because the positive or negative on-axis shear stress makes no difference to the strength of the unit cell [11], some strength parameters should be zero:

F4 ¼ F5 ¼ F6 ¼ 0 F 14 ¼ F 15 ¼ F 16 ¼ 0 F 24 ¼ F 25 ¼ F 26 ¼ 0 F 45 ¼ F 46 ¼ F 56 ¼ 0 Therefore, Eq. (13) can be rewritten as:

F 11 r21 þ F 22 r22 þ F 33 r23 þ F 44 r24 þ F 55 r25 þ F 66 r26 þ 2F 12 r1 r2 þ 2F 13 r1 r3 þ 2F 23 r2 r3 þ F 1 r1 þ F 2 r2 þ F 3 r3 ¼ 1

F 13 ¼ F 23

F 55 ¼ F 66 ¼

Ey ¼

1

;

Ez ¼

S213

here the basic strengths Spqc and Spqt were given in Ref. [11].

Table 1 Mechanical and physical properties of the used materials.

1

S11 S22 S33 1 1 1 Gxy ¼ ; Gyz ¼ ; Gxz ¼ S66 S44 S55 S S S cxy ¼  12 ; cyz ¼  23 ; czx ¼  31 S11 S22 S33

1

ð16Þ

ð11Þ

The engineering constants of 3D braided composites can be written as:

;

1 1 1 1  ; F3 ¼  S11t S11c S33t S33c 1 1 1 ¼ ; F 33 ¼ ; F 44 ¼ 2 ; S11t S11c S33t S33c S12 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 11 F 33 ; F 12 ¼  F 11 F 22 ¼ 2 2

F1 ¼ F2 ¼

Then, the stiffness matrix of a unit cell in Eq. (10) is inverted to the compliance matrix:

1

ð15Þ

where Fp and Fpq are determined as follows:

F 11 ¼ F 22

Ex ¼

ð14Þ

F 34 ¼ F 35 ¼ F 36 ¼ 0

ð10Þ

½S ¼ ½C1

ð13Þ

Type Material

ð12Þ

E33 (GPa) E22 (GPa) G32 (GPa) G12 (GPa) c32

A

Carbon fiber 230 Epoxy resin 3.5

40

24

14.3

0.25

B

Carbon fiber 234.6 Epoxy resin 2.94

13.8

13.8 1.7

5.5

0.2

cm 0.35

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L. Jiang et al. / Composite Structures 100 (2013) 511–516

(a) 100

Present model Fiber inclination model

25

20

Gxz=Gyz (GPa)

80

Ez (GPa)

(d) 30

Measured [27] Present model Fiber inclination model

60

40

15

10

20

5

0

0 10

20

30

40

10

50

20

Braid Angle (Deg.)

(b)

30

(e) 1.0

Present model Fiber inclination model

25

30

40

50

40

50

Braid Angle (Deg.)

Present model Fiber inclination model

0.8

xy

0.6 15

γ

Ex= Ey (GPa)

20

0.4 10 0.2

5

0

0.0 10

20

30

40

50

10

20

Braid Angle (Deg.)

(c)

30

(f)

Present model Fiber inclination model

25

30

Braid Angle (Deg.) 1.0 Present model Fiber inclination model 0.8

0.6

γzx= γ zy

Gxy (GPa)

20

15

0.4

10 0.2

5

0

0.0 10

20

30

40

50

10

20

Braid Angle (Deg.)

30

40

50

Braid Angle (Deg.)

Fig. 3. Effective elastic constants of 3D braided composites.

To determine when failure occurs in the unit cell, the imposed simple longitudinal stress rz upon the unit cell is transformed from the global coordinate system to material principal direction in one of the yarns [26], as follow:

½ r1

T

T

T

r2 r3 r4 r5 r6  ¼ ½T ½ 0 0 rz 0 0 0 

ð17Þ

Considering the curve of braiding yarns in Fig. 2, an amplification factor k for r3 was considered, as follows:

k¼

r3 þ r0 r3

ð18Þ

where r0 is the additional bending stress, which can be calculated:

r0 ¼

Qld d  4Fed 4IP

ð19Þ

here F denotes the internal force of a yarn, e is the distance from the midpoint of a yarn to the chord line oA, d represents the diameter of a yarn, ld is the length of the diagonal of unit cell, IP is the polar mo-

L. Jiang et al. / Composite Structures 100 (2013) 511–516 Table 2 Comparisons of the predicted longitudinal strength and corresponding measured [26] of 3D braided composites. Sample number

Surface braid angle (°)

Fiber volume fraction

Experimental results (MPa)

Theoretical predictions (MPa)

Error (%)

1 2 3 4 5 6

17 17 17 20 25 25

0.38 0.40 0.46 0.46 0.46 0.29

165.0 165.0

175.6 176.8 179.3 174.4 166.9 142.2

6.42 7.15

183.0 120.0

4.70 18.50

ment of inertia of a yarn, Q is the interaction force between the yarns and is calculated according to the deformation compatibility condition.

Q¼

5Fe ld

ð20Þ

Thus, Eq. (15) can be rewritten as:

F 11 r21 þ F 22 r22 þ F 33 ðkr3 Þ2 þ F 44 r24 þ F 55 r25 þ F 66 r26 þ 2F 12 r1 r2 þ 2F 13 r1 ðkr3 Þ þ 2F 23 r2 ðkr3 Þ þ F 1 r1 þ F 2 r2 þ F 3 ðkr3 Þ ¼ 1

515

6. Conclusion In this paper, the effective elastic constants and the longitudinal strength of 3D braided composites are deduced theoretically based on the helix geometry model. The effect of braid angle on the elastic properties and longitudinal strength are investigated. The relationship between the longitudinal strength and fiber volume fraction is also studied. The theoretical calculation results are compared to reported experimental findings in the literature and excellent results are obtained. The following conclusions are drawn from a detailed analysis of the data: (i) The numerical prediction of the present model shows a good agreement with the experimental data compared with the fiber inclination model. (ii) The longitudinal Young’s modulus and shear modulus (Ez, Gxy, Gxz, Gyz) predicted by the present model are higher than the fiber inclination model. (iii) The braid angle has significant influences on the effective elastic constants and longitudinal strength of 3D braided composites. The longitudinal strength increases with the increase of fiber volume fraction.

ð21Þ

Substituting Eqs. (16)–(20) into Eq. (21), the longitudinal failure strength of 3D braided composites under uniaxial load can be determined. 5. Results and discussions In the present work, the effective elastic constants and the longitudinal failure strength of 3D braided composites are predicted using the helix geometry model. The results are then compared with the available corresponding measurements. Two types of braided composites are studied in this work. Type A is used for analyzing the elastic constants of 3D braided composites; type B is used for predicting the strength. The yarn and resin elastic properties are given in Table 1. Fig. 3 shows the effects of braid angle on the engineering elastic constants of 3D braided composites based on the helix geometry model and conventional unit cell model. It is found that the present predictions based on the helix geometry model show a better agreement with the experimental data [27] than the fiber inclination model. The longitudinal Young’s modulus Ez shows an inverse dependency on braid angle because the yarns’ orientation becomes more parallel to the loading axis as the braid angle decreases, as shown in Fig. 3a. Moreover, the values predicted by the helix geometry model are lower than that predicted by the fiber inclination model. The predictions of the transverse Young’s moduli, Ex and Ey, show very little sensitivity to the braid angle and all the values predicted by the two models are almost the same, as shown in Fig. 3b. For the transverse shear modulus Gxy, it is observed that the predictions based on the fiber inclination model are higher than the helix geometry model. From Fig. 3d–f, the helix geometry model predictions for the longitudinal shear modulus Gxz, Gyz, transverse Poisson’s ratio cxy and longitudinal Poisson’s ratio czx, czy are found to be close to the predictions of the fiber inclination model. Table 2 provides the comparisons of the longitudinal failure strength from the experiment [26] and the present prediction with different braid angle and fiber volume fraction. It is found that the predictions from the present model are in good agreement with the corresponding measurements. The longitudinal strength decreases with increasing braid angle and has a 7% reduction when the braid angle varies from 17° to 25°. The longitudinal strength has a 17% increase when the fiber volume fraction increases from 29% to 46% with 25° braid angle.

Acknowledgements The authors would like to thank the National Natural Science Foundation of China (10772064, 10972070, 11272110) and Excellent Youth Foundation of Heilongjiang Scientific Committee (JC200910). Appendix A

1 ðf1  f2  f3  f4 þ f5 þ f6 þ Ln3 þ f7  f8 þ n1 U 2n1 þ n2 V  f9 þ f10  f11 þ f12  f13  2n3  f14   pffiffiffiffiffiffi þf15  4n2 f16 =f17

F 1 ðzÞ ¼

pffiffiffiffiffiffi F 2 ðzÞ ¼ 0:5ðf18 þ 4 f16 Þ=f17

ðA1Þ ðA2Þ

where

f1 ¼

4c2 l2 n21 n2   2 2 2 c3 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

4Ll2 l3 n21 n2  f2 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA3Þ

ðA4Þ

f3 ¼

4m2 l3 n21 n2   2 2 c3 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA5Þ

f4 ¼

4c2 l1 n1 n22   2 2 2 c3 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA6Þ

4Ll1 l3 n1 n22  f5 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22 f6 ¼

4m1 n1 n22   2 2 2 c3 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA7Þ

ðA8Þ

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L. Jiang et al. / Composite Structures 100 (2013) 511–516

4Ll1 l2 n1 n2 n3  f7 ¼  2 2 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA9Þ

2

4Ll1 n22 n3  f8 ¼  2 2 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA10Þ

2

4Vl2 n21 n2  f9 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA11Þ

8Vl1 l2 n1 n22  f10 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA12Þ

2

4Vl1 n32  f11 ¼  2 2 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA13Þ

8zl1 l3 n21 n2  f12 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22

ðA14Þ

8zl1 l3 n1 n22  f13 ¼  2 2 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA15Þ

8zl1 l2 n1 n2 n3  f14 ¼  2 2 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA16Þ

2

8zl1 n22 n3  f15 ¼  2 2 2 4l2 n1  8l1 l2 n1 n2 þ 4l1 n22 2

ðA17Þ

2

f16 ¼ c22 l2 n41  4c1 c3 l2 n41  2c2 l2 m2 n41 þ 2Lc3 l2 l3 m2 n41 2

þ m22 n41  2Lc3 l2 m3 n41  2c22 l1 l2 n31 n2 þ 8c1 c3 l1 l2 n21 n2 þ

2c2 l2 m1 n31 n2

 2Lc3 l2 l3 m1 n31 n2 þ 2c2 l1 m2 n31 n2

 2Lc3 l1 l3 m2 n31 n2  2m1 m2 n31 n2 þ 4Lc3 l1 l2 m3 n31 n2 2

2

þ c22 l1 n21 n22  4c1 c3 l1 n21 n22  2c2 l1 m1 n21 n22 2

þ 2Lc3 l1 l3 m1 n21 n22 þ m21 n21 n22  2Lc3 l1 m3 n21 n22 2

þ 2Lc3 l2 m1 n31 n3  2Lc3 l1 l2 m2 n31 n3  2Lc3 l1 l2 m1 n21 n2 n3 2

2

þ 2Lc3 l1 m2 n21 n2 n3  4c3 l2 l3 m2 n41 z þ 4c3 l2 m3 n41 z þ

4c3 l2 l3 m1 n31 n2 z

þ

4c3 l1 l3 m2 n31 n2 z

 8c3 l1 l2 m3 n31 n2 z

2

2

 4c3 l1 l3 m1 n21 n22 z þ 4c3 l1 m3 n21 n22 z  4c3 l2 m1 n31 n3 z þ 4c3 l1 l2 m2 n31 n3 z þ þ4c3 l1 l2 m1 n21 n2 n3 z 2

 4c3 l1 m2 n21 n2 n3 z

ðA18Þ

  2 2 f17 ¼ c3 4l2 n21  8l1 l2 n1 n2 þ 4l1 n22

ðA19Þ

f18 ¼ 4Lc3 l2 l3 n21  4c2 l2 n21 þ 4m2 n21 þ 4c2 l1 n1 n2  4Lc3 l1 l3 n1 n2  4m1 n1 n2  4Lc3 l1 l3 n1 n3 2

2

 4Lc3 l1 l2 n1 n3 þ 4Lc3 l1 n2 n3 þ 4Vc3 l2 n21  8Vc3 l1 l2 n1 n2 þ

2 4Vc3 l1 n22

þ 8c3 l1 l2 n1 n3 z 

2 8c3 l1 n2 n3 z

 8c3 l2 l3 n21 z þ 8c3 l1 l3 n1 n2 z ðA20Þ

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